Show that 1/(2x+1) - 1/((2x+1)²) + 4/(4x-1) = (24x²+14x+4) / ((2x+1)²(4x-1))

Mark scheme

Marking points from the official mark scheme (Oct/Nov 2022). The mark or mark type is shown in brackets.

1. [1] (a) For attempt to obtain a single fraction (e.g., by multiplying 1/(2x+1) by (2x+1)(4x-1), etc.)
2. [1] (a) Must see sufficient detail of expansion and collecting terms, correct answer shown as given (AG)
3. [1] (b) Integral of 1/(2x+1) is 1/2 ln(2x+1)
4. [1] (b) Integral of -1/((2x+1)²) is 1/(2(2x+1)) or (2x+1)⁻¹ / (2*1) or equivalent
5. [1] (b) Integral of 4/(4x-1) is ln(4x-1)
6. [1] (b) For correct application of limits 0 to 1, with at least one log term and all terms bracketed/manipulated correctly
7. [1] (b) For application of log laws to simplify (e.g. 1/2 ln3 + ln3 - 1/2 ln2 from evaluation)
8. [1] (b) 1/2 ln(27/12)
9. [1] (b) Constant term is 1/2